EURASIP Journal on Advances in Signal ProcessingVolume 2007, Article ID 25672, 13 pages doi:10.1155/2007/25672 Research Article A Generalized Algorithm for Blind Channel Identification w
Trang 1EURASIP Journal on Advances in Signal Processing
Volume 2007, Article ID 25672, 13 pages
doi:10.1155/2007/25672
Research Article
A Generalized Algorithm for Blind Channel Identification with Linear Redundant Precoders
Borching Su and P P Vaidyanathan
Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
Received 25 December 2005; Revised 19 April 2006; Accepted 11 June 2006
Recommended by See-May Phoong
It is well known that redundant filter bank precoders can be used for blind identification as well as equalization of FIR channels Several algorithms have been proposed in the literature exploiting trailing zeros in the transmitter In this paper we propose a generalized algorithm of which the previous algorithms are special cases By carefully choosing system parameters, we can jointly optimize the system performance and computational complexity Both time domain and frequency domain approaches of chan-nel identification algorithms are proposed Simulation results show that the proposed algorithm outperforms the previous ones when the parameters are optimally chosen, especially in time-varying channel environments A new concept of generalized signal richness for vector signals is introduced of which several properties are studied
Copyright © 2007 B Su and P P Vaidyanathan This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 INTRODUCTION
Wireless communication systems often suffer from a
prob-lem due to multipath fading which makes the channels
frequency-selective Channel coefficients are often unknown
to the receiver so that channel identification needs to be done
before equalization can be performed Among techniques
for identifying unknown channel coefficients, blind
meth-ods have long been of great interest In the literature many
blind methods have been proposed based on the knowledge
of second-order statistics (SOS) or higher-order statistics of
the transmitted symbols [1,2] These methods often need to
accumulate a large number of received symbols until
chan-nel coefficients can be estimated accurately This requirement
leads to a disadvantage when the system is working over a
fast-varying channel
A deterministic blind method using redundant filterbank
precoders was proposed by Scaglione et al [3] by exploiting
trailing zeros introduced at the transmitter.Figure 1shows
a typical linear redundant precoded system Source
sym-bols are divided into blocks with size M and linearly
pre-coded intoP-symbol blocks which are then transmitted on
the channel It is well known that whenP ≥ M + L, where
L is the maximum order of the FIR channel, interblock
in-terference (IBI) can be completely eliminated in absence of
noise When the block sizeM increases, the bandwidth
effi-ciencyη =(M + L)/M approaches unity asymptotically The deterministic method proposed in [3] (which we will call the SGB method) exploits trailing zeros with lengthL introduced
in each transmitted block and assumes the input sequence
is rich That is, the matrix composed of finite source blocks
achieves full rank
The method in [3] requires the receiver to accumulate
at leastM blocks before channel coefficients can be
identi-fied This prevents the system from identifying channel co-efficients accurately when the channel is fast-varying, espe-cially when the block sizeM is large More recently,
Man-ton and Neumann pointed out that the channel could be identifiable with only two received blocks [4] An algorithm based on viewing the channel identification problem as find-ing the greatest common divisor (GCD) of two polynomi-als is proposed in [5] (which we will call the MNP method) Eventhough it greatly reduces the number of received blocks needed for channel identification, the algorithm has much more computational complexity especially when the block sizeM is large.
In this paper, we propose a generalized algorithm of which the SGB algorithm proposed in [3] and the MNP al-gorithm in [5] are both special cases By carefully choos-ing parameters, the system performance and computational
Trang 2s1 (n)
s2 (n)
s M(n)
Vector
s(n)
Precoder
u1 (n)
u2 (n)
u P(n)
Equalizer
Channel
P
P
P
P
P
P
e(n)
H(z)
z 1
z 1
z 1
z
z
z
Vector
y(n)
y1 (n)
y2 (n)
y P(n)
s1 (n)
s2 (n)
s M(n)
Vector
s(n)
.
.
.
.
Figure 1: Communication system with redundant filter bank precoders
complexity can be jointly optimized The rest of the paper
is organized as follows.Section 2describes the system
struc-ture with linear precoder filter banks and reviews several
existing blind algorithms InSection 3we present the
gen-eralized algorithm and derive the conditions on the input
sequence under which the algorithm operates properly In
Section 4we propose a frequency domain version of the
gen-eralized algorithm The concept of gengen-eralized signal richness
is introduced inSection 5 and some properties thereof are
studied in detail Simulation results and complexity
analy-sis of both time and frequency domain approaches are
pre-sented in Section 6 In particular, simulations under
time-varying channel environments are presented to demonstrate
the strength of the proposed algorithm against channel
vari-ation Finally, conclusions are made inSection 7 Some of the
results in the paper have been presented at a conference [6]
1.1 Notations
Boldfaced lower-case letters represent column vectors
Bold-faced upper-case letters and calligraphic upper case letters
are reserved for matrices Superscripts as in AT and A†
de-note the transpose and transpose-conjugate operations,
re-spectively, of a matrix or a vector All the vectors and
ma-trices in this paper are complex-valued In the figures “↑ P”
represents an expander and “↓ P” a decimator [7]
If v = [v1 v2 · · · v M T] is an M ×1 column
vec-tor, thenT (v, q) denotes an (M + q −1)× q Toeplitz
ma-trix whose first row and first column are [v1 0 · · · 0] and
[v1 v2 · · · v M 0 · · · 0T], respectively For example,
T
⎛
⎜
⎜
⎡
⎢
⎢
a1
a2
a3
a4
⎤
⎥
⎥, 3
⎞
⎟
⎟
⎠ =
⎡
⎢
⎢
⎢
⎢
a1 0 0
a2 a1 0
a3 a2 a1
a4 a3 a2
0 a4 a3
0 0 a4
⎤
⎥
⎥
⎥
2 PROBLEM FORMULATION AND LITERATURE REVIEW
2.1 Redundant filter bank precoders
Consider the multirate communication system [8] depicted
inFigure 1 The source symbolss1(n), s2(n), , s M(n) may
come fromM different users or from a serial-to-parallel
op-eration on data of a single user For convenience we consider
the blocked version s(n) as indicated The vector s(n) is
pre-coded by aP × M matrix R(z) where P > M The information
with redundancy is then sent over the channelH(z) We
as-sumeH(z) is an FIR channel with a maximum order L, that
is,
H(z) =L
k =0
The signal is corrupted by channel noise e(n) The
re-ceived symbols y(n) are divided into P × 1 block
vec-tors y(n) The M × P matrix G(z) is the channel
equal-izer and s1(n),s2(n), ,s M(n) are the recovered symbol
streams Also, for simplicity we define h as the column vector
[h0 h1 · · · h L ] We set
that is, the redundancy introduced in a block is equal to the maximum channel order
2.2 Trailing zeros as transmitter guard interval
Suppose we choose the precoder R(z) =[R1
0 ] where R1is an
M × M constant invertible matrix and the L × M zero matrix
0 represents zero-padding with lengthL in each transmitted
block, as indicated inFigure 2 For simplicity of describing the algorithms, in this section we assume the noise is absent
Trang 3s1 (n)
s2 (n)
s M(n)
Vector
s(n)
R1
u1 (n)
u2 (n)
u M(n)
Vector
u(n)
Block of
L zeros
Noisee(n)
Channel
P
P
P
P
P
P
P
P
u(n) H(z) y(n)
z 1
z 1
z 1
z 1
z 1
z
z
z
Vector
y(n)
y1 (n)
y2 (n)
y P(n)
.
.
.
.
.
Figure 2: The zero-padding system with precoder R1
Now, the received blocks can be written as
y(1) y(2) · · · y(J)
Y matrix; sizeP × J
=HMR1
s(1) s(2) · · · s(J)
,
S matrix; sizeM × J
(4)
whereHM = T (h, M) is the full-banded Toeplitz channel
matrix As long as vector h is nonzero, the matrixHM has
full column rankM Now, we assume the signal s(n) is rich,
that is, there exists an integerJ such that the matrix S has
full row rank M Since R1 is an M × M invertible matrix,
we conclude that theP × J matrix Y has rank M So there
existL linearly independent vectors that are left annihilators
of Y In other words, there exists aP × L matrix U0such that
U†0Y=UHMR1S=0 Now that R1S has rankM, this implies
The channel coefficients h can then be determined by solving
(5) In practice where channel noise is present, the
computa-tion of the annihilators is replaced with the computacomputa-tion of
the eigenvectors corresponding to the smallestL singular
val-ues of Y In this and the following sections, the channel noise
term is not shown explicitly
Note that this algorithm [3] works under the assumption
that S has full row rankM Obviously J ≥ M is a necessary
condition for this assumption This means the receiver must
accumulate at leastM blocks (i.e., a duration of M(M + L)
symbols) before channel identification can be performed
This could be a disadvantage when the system is working over
a fast-varying channel
2.3 The GCD approach
Another approach proposed in [5] requires only two received blocks for blind channel identification Recall that the
chan-nel is described by y=HMu= T (h, M)u, or
⎡
⎢
⎢
⎣
y1
y2
y P
⎤
⎥
⎥
⎦=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
h1
h L h1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎡
⎢
⎢
⎣
u1
u2
u M
⎤
⎥
⎥
By multiplying [1 x x2 · · · x P −1] to both sides of (6), we obtain
where
y(x)
P−1
k =0
y k+1 x k, h(x)
L
k =0
h k x k,
u(x)
M−1
k =0
u k+1 x k
(8)
are polynomial representations of the output vector, channel vector, and input vector, respectively This means, (6) is noth-ing but a polynomial multiplication Now, suppose we have
two received blocks y(1) and y(2), and lety1(x) = h(x)u1(x)
and y2(x) = h(x)u2(x) represent the polynomial forms of these Then the channel polynomialh(x) can be found as the
GCD of y1(x) and y2(x), given that the input polynomials
u1(x) and u2(x) are coprime to each other.
Trang 4To compute the GCD of y1(x) and y2(x), we first
con-struct a (2P −1)×2P matrix [9]
YP
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
y11 0 · · · 0 y21 0 · · · 0
y12 y11 y22 y21
y12 0 y22 0
y1P . y
21
0 y1P y12 0 y2P y22
.
0 · · · 0 y1P 0 · · · 0 y2P
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
. (9)
One can verify that
YP =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
h1
h L h1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
matrix HM+P −1
size(2P −1)×(M+P −1)
⎡
⎢
⎢
⎢
⎢
⎢
⎢
u12 u22
21
u1M u12 u2M u22
0 u1M 0 u2M
⎤
⎥
⎥
⎥
⎥
⎥
⎥
matrix U size(M+P −1)×2P
.
(10) Whenu1(x) and u2(x) are coprime to each other, it can
be shown that the matrix U has full rank M + P −1 (see
Section 5) SinceHM+P −1 = T (h, M + P −1) also has rank
M + P −1, rank(YP)= M + P −1 and hence YP hasL left
annihilators (i.e., there exists a (2P −1)× L matrix U0such
that U†0Y = 0) These annihilators are also annihilators of
each column of matrixHM+P −1, and we can therefore, in
ab-sence of noise, identify channel coefficients h0,h1, , h Lup
to a scalar ambiguity In presence of noise, the columns of
U0would be selected as the eigenvectors associated with the
smallest singular values of YP
2.4 Connection to the earlier literature
The MNP method described above can be viewed as a dual
version of the subspace methods proposed in the earlier
lit-erature in multichannel blind identification [10,11] In the
subspace method in [11], the single source can be estimated
as the GCD of the received data from two (more generallyN)
different antennas The MNP method [5] swaps the roles of
data blocks and multichannel coefficients
3 A GENERALIZED ALGORITHM
In this section we propose a generalized algorithm of which
each of the two algorithms described in the previous section
is a special case Comparing the two algorithms described
above, we find that the MNP approach needs much fewer
received blocks for blind identifiability However, it has more
computational complexity Each received block is repeatedP
times to build a big matrix Using the generalized algorithm,
we can choose the number of repetitions and the number of
received blocks freely as long as they satisfy a certain con-straint
3.1 Algorithm description
Observe (6) again and note that it can be rewritten as
T (y, Q) = T (h, M + Q −1)T (u, Q), (11) whereT (·,·) is defined as in (1) HereQ can be any positive
integer Note that in the MNP methodQ is chosen as P, as
described in the previous section Suppose the receiver gath-ersJ blocks with J ≥ 2 Then we have Y(Q J) =HM+Q −1U(Q J), where
Y(Q J) =Ty(1),Q
Ty(2),Q
· · · Ty(J), Q
,
HM+Q −1= T (h, M + Q −1),
(12)
U(Q J) =Tu(1),P
· · · Tu(J), P
Note that U(Q J) has size (M + Q −1)× QJ and Y(Q J)has size (P + Q −1)× QJ For notational simplicity, from now on we
will use subscriptQ as in N Qto denoteN Q = N +Q −1 where
N denotes a positive integer In particular,
M Q = M + Q −1,
Notice that they still have the relationshipP Q = M Q+L.
Assume now the matrix U(Q J)has full row rankM Q Taking
singular-value decomposition (SVD) of Y(Q J)we have
Y(Q J) =Ur U0
Σ 0
Vr V0
†
The size ofΣ is M Q × M Qsince bothHM Q and U(Q J)have full rankM Q The columns of theM Q × L matrix U0are left
an-nihilators of matrix Y(J)and also of H since U(J)has full row rank Suppose
U†0=
⎡
⎢
⎢
⎣
u11 u12 · · · u1,P+Q −1
u21 u22 · · · u2,P+Q −1
u L1 u L2 · · · u L,P+Q −1
⎤
⎥
⎥
Form the Hankel matrices
Uk
⎡
⎢
⎢
⎣
u k1 u k2 · · · u k,L+1
u k2 u k3 · · · u k,L+2
u k,M Q u k,M Q+1 · · · u k,P Q
⎤
⎥
⎥
fork, 1 ≤ k ≤ L Then we have
⎡
⎢
⎢
⎣
U1
U2
UL
⎤
⎥
⎥
⎦
U matrix; size LM Q ×(L+1)
Vector h can thus be identified up to a scalar ambiguity.
Trang 5v(n)
Vector
vQ(n)
N
N
N Q N Q
N Q
N Q
z 1
z 1
Q
Q 1
⎡
⎣IN
0
⎤
⎦
1
Q 2
⎡
⎢
⎣
0
IN 0
⎤
⎥
⎦
Q 1
⎡
⎣0
IN
⎤
⎦
.
.
.
Figure 3:Q-repetition and shifting operation.
3.2 Q-repetition and shifting operation
As we can see in the previous section, the repetition and
shifting operation on a vector signal is crucial in the
gener-alized algorithm.Figure 3gives a block diagram of this
oper-ation For future notational convenience, the subscriptQ as
in vQ(n) denotes the result of this operation on a vector
sig-nal By viewing (11) and applying this operation on y(n) and
u(n), we obtain the relationship
yQ(n) =HM+Q −1uQ(n)
for any positive integerQ.
3.3 Special cases of the algorithm
The blind channel identification algorithm described above
uses two parameters: (a) the number of received blocksJ; (b)
the number of repetitions per blockQ A number of points
should be noted here:
(1) the algorithm works for anyJ and Q as long as U(Q J)has
full row rankM Q This is the only constraint for choosing
parameters J and Q;
(2) note that if we choose Q = 1 andJ ≥ M, then the
algorithm reduces to the SGB algorithm [3];
(3) if we chooseQ = P and J = 2, it becomes the MNP
algorithm [5]
So both the SGB method and the MNP method are a
special case of the proposed algorithm Since U(Q J) has size
M Q × QJ, U(Q J) having full row rank implies QJ ≥ M Q =
M + Q −1, or
Q ≥ M −1
Also note that we cannot chooseJ =1 since U(Q J)can never
have full rank unless the block sizeM =1 This is consistent
with the theory that two blocks are required for blind
chan-nel identification [4] While the inequality (19) is a necessary
condition for U(Q J)to have full rank, it is not sufficient because
it also depends on the values of entries of u(n) Nevertheless,
when inequality (19) is satisfied, the probability of U(Q J) hav-ing full rank is usually close to unity in practice, especially when a large symbol constellation is used Thus,
Q =
M −1
J −1
(20) appears to be a selection that minimizes the computational cost given the number of received blocksJ A detailed study
on the conditions for U(Q J) to have full rank is presented in
Section 5 WhenJ =2,Q can be chosen as small as M −1 rather thanP If we take J =3,Q = (M −1/2) makes the matrix
Y twice smaller We can chooseQ = 1 only whenJ ≥ M.
This coincides with the SGB algorithm which uses a richness assumption [3]
4 FREQUENCY DOMAIN APPROACH
In this section we slightly modify the blind identification al-gorithm and directly estimate the frequency responses of the channel at different frequency bins and equalize the channel
in the frequency domain We call the modified algorithm fre-quency domain approach Some of the ideas come from [12] The receiver structure for the frequency domain approach is shown inFigure 4 To demonstrate how this system works, observe theP Q × M Q full-banded Toeplitz channel matrix
HM Q =Th,M Q
Define a row vector vT
ρ = [1 ρ −1 · · · ρ −(P Q −1)] withρ a
nonzero complex number Due to full-banded Toeplitz struc-ture ofHM Q, we have
vT
ρHM Q =H(ρ) ρ −1H(ρ) · · · ρ −(M Q −1)H(ρ)
, (22) whereH(ρ) = L
k =0h k ρ − kis the channelz-transform
evalu-ated atz = ρ.
LetN be chosen as an integer greater than or equal to P Q, and letρ1,ρ2, , ρ N be distinct nonzero complex numbers Consider anN × P Qmatrix VN × P Qwhoseith row is v T
ρ i:
VN × P Q =
⎡
⎢
⎢
⎢
⎢
1 ρ −1 ρ −2 · · · ρ −(P Q −1)
1
1 ρ −1 ρ −2 · · · ρ −(P Q −1)
2
1 ρ −1
N ρ −2
N · · · ρ −(P Q −1)
N
⎤
⎥
⎥
⎥
⎥. (23)
It is easy to verify that
VN × P QHM Q =ΛN
⎡
⎢
⎢
⎢
⎢
1 ρ −1 · · · ρ −(M Q −1)
1
1 ρ −1 · · · ρ −(M Q −1)
2
1 ρ −1
N · · · ρ −(M Q −1)
N
⎤
⎥
⎥
⎥
⎥,
VN × MQmatrix
(24)
Trang 6and shifting V NP Q
P
P
P
y(n)
z
z
z
Vector
y(n)
y1 (n)
y2 (n)
y P(n)
Vector
yQ(n)
y Q1(n)
y Q2(n)
y QP Q(n)
Vector
z(n)
z1 (n)
z2 (n)
z N(n)
.
.
Figure 4: Receiver structure for frequency domain approach
where
ΛN =diag
H
ρ1
H
ρ2
· · · H
ρ N
diag hN
(25)
is a diagonal matrix with frequency domain channel coe
ffi-cients as the diagonal entries Now, when we gather receiving
blocks and repeat them as in (12), we get the following
ma-trix:
Y(Q J) =T (y(1), Q) T (y(2), Q) · · · Ty(J), Q
(26)
Since we have Y(Q J) = HM QU(Q J) in absence of noise, by
multiplying VN × P Q and Y(Q J), we have
Z=VN × P QY(Q J) =VN × P QHM QU(Q J) =ΛNVN × M QU(Q J) (27)
Recall that rank(Y(Q J))=rank(U(Q J))= M Q Sinceρ1,ρ2, , ρ N
are all distinct, the matrix Z has the same rank as Y(Q J) The
dimension of the null space of matrix Z is henceN − M Q By
performing SVD on Z, we can find theseN − M Q left
anni-hilators of Z, which are also annianni-hilators of ΛNVN × M Q There
exists an (N− M Q)× N matrix U †0 such that U†0Z=0 Since
U(Q J)has full rank, this implies
U†0ΛNVN × M Q =0. (28) Suppose
U†0=
⎡
⎢
⎢
⎣
u11 u12 · · · u1N
u21 u22 · · · u2N
u N − M Q,1 u N − M Q,2 · · · u N − M Q,N
⎤
⎥
⎥
⎦. (29)
Then by observing thei jth entry of (28), we have
u† i jh †
for alli, j, 1 ≤ i ≤ N − M Q and 1≤ j ≤ M Q, where ui j =
[u i1 ρ1−(j −1) u i2 ρ2−(j −1) · · · u iN ρ − N(j −1)]† Hereh Nis the row
vector in (25) Form theM Q × N matrices
Ui =
⎡
⎢
⎢
⎢
⎢
u i1 ρ −1 u i2 ρ −1 · · · u iN ρ −1
N
u i1 ρ −2 u i2 ρ −2 · · · u iN ρ −2
N
u i1 ρ −(M Q −1)
1 u i2 ρ −(M Q −1)
2 · · · u iN ρ −(M Q −1)
N
⎤
⎥
⎥
⎥
⎥,
(31) and letU =[UT
2 · · · UT
N − M Q]T Then from (30) we haveUh N = 0 Then the frequency domain channel
coeffi-cientsh N can be estimated by solving this equation After the
frequency domain channel coefficients are estimated, the re-ceived symbols can be equalized directly in the frequency do-main, as in DMT systems
Recall that we have the freedom to chooseN as any
inte-ger greater than or equal toP Qand the values ofρ i, 1≤ i ≤ N
as any nonzero complex number in thez-domain In this
pa-per, we useN = P Qand
ρ k =exp j2kπ
N
!
, k =0, 1, , N −1. (32) Note that sinceH(z) is an Lth order system, there are
at mostL values among H(ρ i) which can be zero (channel nulls) By choosingN ≥ P Q, there are at leastM Q nonzero values among H(ρ i), i = 1, 2, , P Q In practice we can choose to equalize the received symbols in frequency bins as-sociated with the largest M Q frequency responses H(ρ i) to enhance the system performance This provides resistance to channel nulls
5 GENERALIZED SIGNAL RICHNESS
For the generalized blind channel identification method
pro-posed in this paper to work properly, the matrix U(Q J) de-fined in (13) must have full row rank for given parame-tersJ and Q An obvious necessary condition has been
pre-sented as inequality (19) inSection 3 The sufficiency,
how-ever, depends on the content of signal u(n) When Q =
1 and u(n) is rich, then there exists J such that U(Q J) =
[u(0) u(1) · · · u(J − 1)] has full rank When Q > 1, u(n)
requires another kind of richness property so that U(Q J) has full rank for a finite integerJ We call this property the gener-alized signal richness and define it as follows.
Definition 1 An M ×1 sequence u(n), n ≥ 0 is said to be (1/Q)-rich if there exists a finite integer J such that the (M +
Q −1)× JQ matrix
U(Q J) =Ts(0),Q
Ts(1),Q
· · · Ts(J), Q
(33) has full row rankM + Q −1
Several interesting properties of generalized signal rich-ness will be presented in this section The reason why we use the notation of (1/Q) will soon be clear when these
proper-ties are presented
Trang 75.1 Measure of generalized signal richness
Lemma 1 If an M × 1 sequence s( n) is (1/Q)-rich, then s(n)
is (1 /(Q + 1))-rich.
Proof See the appendix.
Lemma 1 states a basic property of generalized signal
richness: the smaller the value ofQ is, the “stronger” the
con-dition of (1/Q)-richness is For example, if an M ×1 sequence
s(n) is 1-rich, or simply rich, then it is (1/Q)-rich for any
pos-itive integerQ On the contrary, a (1/2)-rich signal s(n) is not
necessarily 1-rich We can thus define a measure of
general-ized signal richness for a givenM ×1 sequence s(n) as follows.
Definition 2 Given an M ×1 sequence s(n), n ≥ 0, the degree
of nonrichness of s( n) is defined as
Qmin min
Q s(n) is 1
Q-rich
!
Recall that the larger the degree of nonrichnessQminis,
the weaker the richness of the signal s(n) is If s(n) is not
(1/Q)-rich for any Q, then Qmin= ∞ The property of an
in-finite degree of nonrichness can be described in the
follow-ing lemma We use the notation pM(x) to denote the column
vector:
pM(x) =1 x x2 · · · x M −1T
Lemma 2 Consider an M × 1 sequence s( n) The following
statements are equivalent:
(1) s(n) is not (1/Q)-rich for any Q;
(2) the degree of nonrichness of s( n) is infinity;
(3) either there exists a complex number α such that
[1 α · · · α M −1] is an annihilator of s( n) or
[0 · · · 0 1] is an annihilator of s(n);
(4) either polynomials p n(x) =pT M(x)s(n), n ≥ 0 share
a common zero (at α) or their orders are all less than
M − 1.
Proof See the appendix.
Note that the statement [0 · · · 0 1] is an annihilator
of s(n) in condition (3) and the statement that polynomials
p n(x) have orders less than M −1 in condition (4) can be
interpreted as the special situation when the common zeroα
is at infinity
If anM ×1 sequence s(n) has a finite degree of
non-richness, or s(n) is (1/Q)-rich for some integer Q, then it can
be shown that the maximum possible value ofQminisM −1,
as described in the following lemma
Lemma 3 If M > 1 and an M × 1 sequence s( n) is not (1/(M −
1))-rich, then it is not (1/Q)-rich for any Q.
Proof See the appendix.
WithLemma 3, we can see that for anM ×1 sequence
s(n), the possible values of the degree of non-richness Qmin
are 1, 2, , M −1, and ∞ (1/(M −1))-richness is thus the weakest form of generalized richness When using the MNP method [9], this weakest form of generalized richness
is very crucial If this weakest form of richness of s(n) is
not achieved, then by Lemma 2s(n) has an infinite degree
of non-richness and polynomials pT M(x)s(n) have a common
factor (x − α) Then as inSection 2.3, when we take GCD of the polynomials representing the received blocks, the receiver would be unable to determine whether the factor (x − α)
be-longs to the channel polynomial or is a common factor of the
symbol polynomials Therefore, if the input signal s( n) has in-finite degree of non-richness, all methods proposed in this paper will fail for all Q.
Furthermore, the MNP method proposed in [5] usesQ =
P UsingLemma 3, we see that usingQ = M −1 is sufficient
if we are computing the GCD of polynomials representing received blocks and the following two conditions are true: (1) the GCD is known to have a degree less than or equal toL; (2)
the degree of each symbol polynomial is less than or equal to
M −1 UsingQ = P not only is computationally unnecessary,
but also, as we will see in simulation results inSection 6, has sometimes a worse performance than usingQ = M −1 in presence of noise
The sufficiency of Q = M −1 can also be understood from the point of view of polynomial theory Suppose polynomials
a(x) and b(x) have degrees less than or equal to P −1 and have a greatest common denominatord(x) whose degree is
less than or equal toL Suppose a(x) = d(x)a1(x) and b(x) =
d(x)b1(x) and both a1(x) and b1(x) have degrees less than or
equal toM −1 and they are coprime to each other Then there exists polynomialsp(x) and q(x) whose degree are less than
or equal toM −2 such that 1= p(x)a1(x) + q(x)b1(x) and thusd(x) = p(x)a(x) + q(x)b(x).
5.2 Connection to earlier literature
An earlier proposition mathematically equivalent toLemma
3 has been presented in the single-input-multiple-output (SIMO) blind equalization literature [10,13] We review it here briefly
Proposition 1 Let h[ n] be J × 1 vectors Suppose a QJ ×(Q +
M − 1) block Toeplitz matrix
TQ(h)
=
⎡
⎢
⎢
⎢
h[0] h[1] · · · h[M −1] 0 · · · 0
0 h[0] h[1] · · · h[M −1] .
⎤
⎥
⎥
⎥
(36)
satisfies the following conditions:
(1) h[0] = 0 and h[ M −1] = 0;
(2) h[n] = 0 for n < 0 and n ≥ M;
(3) Q ≥ M − 1.
Trang 8ThenTQ (h) has full column rank if and only if
h(z)
M
i =0
h[i]z − i =0, ∀ z. (37)
Here h[n] was used to refer to the impulse response of
aJ ×1 channel.Q stands for the observation period in the
multiple-channel receiver end Conditions (1) and (2) imply
that the channel has finite impulse response Condition (3)
can be met by increasing the observation periodQ While this
old proposition focuses on the coefficients of multiple
chan-nels rather than values of transmitted symbols, it is
mathe-matically equivalent to the statement that s(n) is (1/(M −
1))-rich if and only if polynomials pT M(x)s(n) do not share
com-mon zeros The case ofQ < M −1, however, has not been
considered earlier in the literature, to the best of our
knowl-edge
5.3 Remarks on generalized signal richness
In this section we introduced the concept of generalized
sig-nal richness Given anM ×1 signal s(n), n ≥ 0, the degree
of non-richness Qminwas defined For an input signal with a
degree of non-richnessQmin, we can choose any
and some finiteJ for the generalized algorithm proposed in
Section 3to work properly The possible values ofQminare
1, 2, , M −1, and∞ If s(n) has an infinite degree of
non-richness, the algorithm proposed in this paper will fail for
all Q The degree of non-richness of a signal s(n) directly
depends on its content A deeper study of degree of
non-richness will be presented elsewhere [14]
6 SIMULATIONS AND DISCUSSIONS
In this section, several simulation results, comparisons, and
discussions will be presented We will first test our proposed
method and compare it with the existing methods [3,5]
de-scribed inSection 2 Secondly, we will compare the
perfor-mances of time domain versus frequency domain approaches
and show that under some channel conditions the frequency
domain approach outperforms the time domain approach
Finally, we will analyze and compare the computational
com-plexity of algorithms proposed in this paper
6.1 Simulations of time domain approaches
A Rayleigh fading channel of orderL = 4 is used The size
of transmitted blocks isM =8 and received block size isP =
M+L =12 The normalized least squared channel estimation
error, denoted asE ch, is used as the figure of merit for channel
identification and is defined as follows:
E ch = h−h 2
45 40 35 30 25 20 15 10
SNR (dB)
10 5
10 4
10 3
10 2
10 1
10 0
10 1
M =8;L =4
J =2,Q =12 (GCD)
J =2,Q =1 (SGB)
J =2,Q =8
J =10,Q =12 (GCD)
J =10,Q =1 (SGB)
J =10,Q =2
Figure 5: Normalized least squared channel error estimation
45 40 35 30 25 20 15 10
SNR (dB)
10 7
10 6
10 5
10 4
10 3
10 2
10 1
10 0
M =8;L =4
J =2,Q =12 (GCD)
J =2,Q =1 (SGB)
J =2,Q =8
J =10,Q =12 (GCD)
J =10,Q =1 (SGB)
J =10,Q =2
Figure 6: Bit error rate
whereh and h are the estimated and the true channel vec-
tors, respectively The simulated normalized channel estima-tion error is shown inFigure 5and the corresponding BER is presented inFigure 6 When the number of blocksJ =10, the MNP method (with the number of block repetitionsQ =12) outperforms the SGB method (Q = 1) by a considerable range TakingQ =2 saves a lot of computation and yet yields
a good performance as indicated Furthermore, in the case
ofJ =2, the system withQ =8 even outperforms the orig-inal MNP method withQ = 12 This also strengthens our argument inSection 5that choosingQ as large as P is
unnec-essary
Trang 945 40 35 30 25 20 15
10
SNR (dB)
10 5
10 4
10 3
10 2
10 1
10 0
M =8;L =4
FD 9 blocksQ =1
TD 9 blocksQ =1
FD 9 blocksQ =2
TD 9 blocksQ =2
Figure 7: Normalized least squared channel error estimation
6.2 Simulations of frequency domain approaches
Figure 7 shows the comparison of frequency domain
ap-proach and time domain apap-proach under the channel coe
ffi-cientsH(z) =1− jz −1+ (−1 + 0.01 j)z −2+ (0.01 + j)z −3−
0.01 jz−4
For frequency domain approach, the normalized least
squared channel error is defined as
E ch = h− h 2
where
h=H
ρ1
H
ρ2
· · · H
ρ N
(41)
andh is the estimation of h Simulation results show that
frequency domain approach outperforms time domain
ap-proach especially when the noise level is high While the
fre-quency domain approach does not in general beat the time
domain approach for a random channel, it has been
consis-tently observed that frequency domain approach performs
better than time domain approach when the last channel
co-efficient h(L) has a small magnitude (i.e., at least one zero of
H(z) is close to the origin).
Since we have the freedom to choose values of coefficients
ρ i, the receiver can adjust ρ i dynamically according to the
a priori knowledge of the approximated channel zero
loca-tions This is especially useful when the channel coefficients
are changing slowly from block to block
6.3 Complexity analysis
For the algorithms presented inSection 3, the SVD computa-tion dominates the computacomputa-tional complexity The number
of blocksJ, the number of repetitions per block Q, and the
received block sizeP decide the size of the matrix on which
SVD is taken The complexity of SVD operation on ann × m
matrix [15] is on the order ofO(mn2) withm ≥ n Since Y(Q J)
has size (P +Q −1)× QJ, the complexity is O(QJ(P +Q −1)2)
We can see that the complexity can be greatly reduced by choosing a smallerQ Recall that the SGB method [3] uses
Q =1 and the MNP method [5] usesQ = P We thus have
the following arguments:
(i) the MNP method has a complexity around 4P times
the complexity of the SGB method for anyJ A choice
ofQ between 1 and P could be seen as a compromise
between system performance and complexity; (ii) when J is large, we have the freedom to choose a
smallerQ, as explained in the previous section.
For the frequency domain approach presented inSection 4,
an additional matrix multiplication is required This de-mands extra computational complexity of the order of
O(JP2
Q) However, if the values ρ i are chosen as equally spaced on the unit circle, an FFT algorithm can be ex-ploited and the computational complexity will be reduced to
O(JP QlogP Q) and is negligible compared to the complexity
of SVD operations
6.4 Simulations for time-varying channels
In this section, we demonstrate the capability of the proposed generalized blind identification algorithm in time-varying channels environments The received symbols can be ex-pressed as
y(n) = L
k =0
h(n, k)x(n − k), (42)
where the (L + 1)-tap channel coefficients h(n, k) vary as the
time indexn changes We generate the channel coefficients
as follows During a time intervalT, the channel coefficients
change fromh1(k) to h2(k), where h1(k) and h2(k), 0 ≤ k ≤
L represent two sets of (L + 1)-tap independent coefficients.
The variation of the coefficient is done by linear interpolation such that
h(n, k) =
⎧
⎪
⎪
⎪
⎪
T − n
T h1(k) +
n
T h2(k) otherwise.
(43)
In our simulation, we chooseT =180 Coefficients of h1(k)
andh2(k) are given inTable 1 The size of transmitted blocks
isM =8 and received block size isP = M + L =12 (so the channel coefficients completely change after 15 blocks) Sim-ulations are performed under different choices of J and Q, as indicated in Figures8and9 The normalized least squared
Trang 10Table 1: Coefficients for the time-varying channel.
channel error is defined as
E ch = h−h 2
whereh is the estimated channel and h is the averaged coef-
ficients during the time the channel is being estimated:
h= 1
JP
n0 +JP −1
n = n0
h(n, 0) h(n, 1) · · · h(n, L)T
. (45)
InFigure 8we see that whenJ = 10, the time range is too
large for the algorithm to estimate the time-varying
chan-nel accurately The performance forJ =2 is much better in
high SNR region because the channel does not vary too much
during the time of two blocks However, in low SNR region
the performance forJ =2 becomes bad The case forJ =4
has the best performance among all other choices because the
channel does not vary too much during the duration of four
receiving blocks, and more data are available for accurate
es-timation This simulation result provides clues about how we
can choose the optimalJ: if the channel variation is fast (T is
smaller) we need a smallerJ while we can use a larger J when
T is larger.
6.5 Remarks on choosing the optimal parameters
According to the simulations results above, we summarize
here a general guideline to choose a set of optimal
param-eters in practice
(1) When the channel is constant and for a fixedQ, a larger
J appears to have a better performance (as shown in
Figure 5) since more data are available for accurate
es-timation
(2) When the channel is time-varying, the optimal choice
ofJ depends on the speed of channel variation
Sim-ulation results in Figures 8 and9 suggest when the
channel coefficients completely change in N blocks, a
choice ofJ ≈ N/4 could be appropriate.
(3) SupposeJ is given, a choice of Q as the smallest
inte-ger that satisfies inequality (19) often has a satisfactory
performance A slightly largerQ can sometimes be
bet-ter (seeFigure 5forJ =10) at the expense of a slightly
increased complexity However, if Q is too large, the
performance could be even worse (seeFigure 5forJ =
2,Q =12)
The guidelines above are given by observing the simulation
results An analytically optimal set ofJ and Q is still under
investigation
45 40 35 30 25 20 15 10
SNR (dB)
10 2
10 1
10 0
10 1
M =8;L =4
J =2;Q =8
J =4;Q =3
J =6;Q =2
J =8;Q =2
J =10;Q =2
J =10;Q =1
Figure 8: Normalized channel MSE performance for a time-varying channel
45 40 35 30 25 20 15 10
SNR (dB)
10 3
10 2
10 1
10 0
M =8;L =4
J =2;Q =8
J =4;Q =3
J =6;Q =2
J =8;Q =2
J =10;Q =2
J =10;Q =1
Figure 9: Bit error rate performance for a time-varying channel
6.6 Noise handling for large J
It should be noted that whenJ is very large (and Q =1), the proposed method behaves like a traditional subspace method using second-order statistics Suppose
Y(J) =HU(J)+ E(J), (46)